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28Elk population spatial dynamics investigation
Sorokin P.A. ([email protected]), Morales J.M.
Center of Open Systems and High Technologies of MIPT
Introduction
Individual-based modeling is actively developing direction in biological modeling. Individual-based model (further referred to as IBM) takes into account species interactions with environment and inter-species interactions on individual specie level. IBMs are powerful tools for population's investigation, they allow taking into account and applying wide experimental data related to population. Refer to publication [1] for details on individual-based approach.
In this work we describe the application of individual-based modeling of ecological and biological systems for investigation of elk spatial population dynamics (Cervus canadensis). We describe the IBM of elk population. A number of IBMs of elk and other ungulate populations have been developed ([2] - [6]). A characteristic feature of our model is application of correlated random walk (CRW) mechanism to describe elk movement. CRW mechanism uses stochastic approach to the choice of movement direction; the corresponding distribution depends on landscape factors. Recent analyses of animal movement data has focused on distance moved or movement rate or on summary properties of trajectories ([30], [32] - [36]). Using both step length and turning angles is likely to be more powerful than using just one variable.
The model was developed within the frame of research program being carried out by University of Connecticut (USA), supported by grant NSF 0078130 "Building a mechanistic basis for landscape ecology of ungulate populations". The director of the research program is Prof. Peter V. Turchin.
1. Elk population.
Elk population has been selected for modeling (Cervus canadensis). At the time
of European arrival, the elk was one of the commonest large mammals in North America. Several subspecies were recognized, of which 2 are now extinct, 2 are fairly uncommon, and 1 is widespread in North America (the Rocky Mountain subspecies). The rapid decline in numbers across the historic range was associated with intense hunting, range deterioration, and competition with livestock [7].
The male elk body mass can exceed 500 kg. Females typically range from 50 to 80% of male body mass (approx. 300 kg). Much like other species their size, elk can be long-lived (15 years or so), particularly females.
As one might expect of a species with a cosmopolitan geographic distribution, elk have quite plastic behavioral characteristics. For example, some elk and red deer populations are seasonally migratory, whereas other populations are resident throughout the year ([8], [9]).
Elk are intermediate feeders, often selecting a mix of grasses, sedges, forbs, herbs, lichen, and tree leaves and bark. Diets vary enormously from one location to another and from one season to the next within the same location. On a seasonal basis, there is a tendency for diets to be predominated by higher quality foods (such as forbs, clover, sedges, or grasses) during the spring and summer growing periods, shifting to lower quality foods (such as broad-leafed plants, twigs, and bark) during the fall and winter. High rates of energy and nutrient intake during the growing season make up for deficits incurred during the winter, when fat and muscle catabolism supply much of the energetic requirement, particularly for pregnant females. Experimental studies using manipulated grass swards and captive animals suggest that elk are capable of micro-habitat selection to maximize rates of energy or nutrient gain ([10], [11]).
2. Goal
Ecologists increasingly appreciate the importance of spatial structure in population, community, and ecosystem dynamics ([21], [22], [23], [24], [25], [26], [27], [28]). Understanding the mechanisms underlying spatial patterning of
populations, in particular dispersal, heterogeneity, and spatial scale, is an important research program for both basic and applied ecology. Individual-based approach combined with novel information technologies (such as GPS, GIS, powerful computers) gives new opportunities for investigations in this direction.
One of the important problems that cannot be solved without such understanding is reestablishing elk populations in their former range. There is considerable public interest to this task. Through federal and state conservation agencies, as well as nongovernmental organizations like the Rocky Mountain Elk Foundation, elk have been reintroduced to extirpated areas of at least 11 U.S. states and 3 Canadian provinces, with future reintroductions being considered in New York and Missouri. These reintroduction efforts provide unique opportunities to study the mechanisms of population spread.
The unstated assumption of many current elk reintroduction efforts is that once the reintroduced populations are established, they will spread through the landscape on their own. But this is not obviously true. Yet we do not have a thorough understanding what will happen to these populations after they become established. Will it be necessary to translocate individuals to ensure the spread, or will these populations spread on their own? If the latter, how long will it take? Clearly, movement is the key mechanism for population spread.
As a result of literature search related to elk movement performed by University of Connecticut, much information on home range size, seasonal migration, and habitat use was found, but only 5 published papers report about quantitative movement data ([12], [13], [14], [15], [16]).
In this research we concentrated on investigation of landscape heterogeneity influence to individual elks movements with time scale 1 day and spatial scale hundreds meters. This task is a part of complex research program conducted by University of Connecticut and supported by grant NSF 0078130 "Building a mechanistic basis for landscape ecology of ungulate populations".
3. Individual-based model of elk population
3.1 Brief description
Elk activities directly described by the model:
1. Movement
2. Feeding
Factors taken into account while choosing the movement direction:
1. Feeding conditions
2. Tendency to maintain movement direction Individual properties found based on energy balance equations. Energy conversion processes taken into account:
1. Basic energy expenses (metabolic processes, feeding, etc.)
2. Movement energy expenses
3. Energy gained from food
Basic movement model is correlated random walk (CRW). Individuals are constantly moving, accelerating or slowing down depending on local environment conditions. The movement direction is selected based on stochastic approach; the directions distribution ensures higher probability of selecting direction towards better environment conditions. Distribution parameters (mean value and dispersion) depend on several factors. One of them is a tendency to maintain movement direction; it defines the correlated nature of the random walk. There are landscape features that are attracting the individual while others are repelling.
3.2 Landscape properties
Landscape is modeled by rectangular grid of M x N cells.
Cell properties
(m, n), 0 < m < M, 0 < n < N - cell coordinates Al - cell linear size Environment factors:
Vi - forage biomass density (food quantity) NDFj - neutral detergent fiber (food quality)
mi - minimum forage biomass density U 0 - plant growth for biomass near zero
3.3 Individual properties
(x, y), 0 < x < M, 0 < y < N - cell coordinates (where individual is located)
W - body biomass
Fat - fat biomass
3.4 Movement
The model traces individual movement with time step At. At every step direction
9 and speed s is selected. New position is calculated according to formulas:
n \ fxt+i = xt + cos(9)sAt / Al i yt+1 = yt + sin(9)sAt / Al
The precision for individual location is linear cell size Al, it is included in formulas (1), as individual coordinates are dimensionless values (they define the cell where individual is located).
3.4.1 Movement direction
Direction is selected based on correlated random walk mechanism ([30], [24],
[29]).
At every model step individual selects the movement direction 9, estimating the properties of each cell inside perception radius Rp . For each landscape cell and each
environmental factor, we measure the strength of a directional bias towards or away from that point of attraction/repulsion. This is done by calculating the dispersion parameter of a wrapped Cauchy density functions, C(9 \ju,p),
(2) C (9|j, P) = -1---2 1 ~ P2 9--, 0 <9< 2n, 0 <p< 1
v 7 2n 1 + p - 2 p cos(9 - jj)
that determines the probability of choosing direction 9, given the mean direction j and the dispersion parameter . Here, j corresponds to the compass direction that leads from the individual's current position to the center of the landscape cell being considered, for repulsion forces, it is the direction that lead away from the point of
repulsion; ¡1 = ¡0 + n, jliq - direction towards repulsion point. For p close to 1, the angular variance is very small and the probability density is concentrated around the mean direction ¡.. For p close to 0, the distribution tends to be uniform in the circle.
The value of p for cell i and environmental factor k is related to the local environmental conditions at the landscape cell and to the distance between the individual and the landscape cell as follows:
(3) pi,k = tanh [ak |4,k\Rfk ], 0 < p,k < 1
where Aikk is the attraction strength of point i for factor k, R - is the distance to the cell, ak and ¡3K- parameters (tanh is the hyperbolic tangent).
For some factors attraction/repulsion decreases with distance, in this case ¡3K < 0, and p decreases with distance from 1 at R = 0 to 0 at R
For some factors attraction/repulsion increases with distance, in this case ¡3K > 0, and p increases with distance from 0 at R = 0 to 1 at R
For each factor k we calculate kernel Kk, as the product of the wrapped Cauchy density functions for all the landscape cells within the perception radius Rp . For
computational purposes we do this for a discrete number, N, selected with equal step.
(4) Kk (e] ) = n C (0J\01, pi.k) for 0, in {1x N x N)
where Kk(0) - directional kernel for factor k, calculated as the product of n Wrapped Cauchy density functions (one for each landscape cell).
The direction persistence is calculated in a slightly different way:
(5) Kc (0 ) = C (0 \ 0p, pCF) C (0 \ 0p - n, pCR),
where 0p - is the direction of the previous movement, pcF is a measure of the strength of the autocorrelation for movement in the same direction, and pcR - for movement in the opposite direction. Once all the kernels for each type of factor have been calculated, a composite movement kernel is calculated by multiplying the normalized kernels:
(6) K(0 ) = KC 0 )nKk (0,)
k=1
where H is the total number of factors considered. The composite kernel is then normalized and probabilities of movement for all directions are calculated:
N
(7) p = Factor • K (0), £ P = 1
i=1
where Factor is a normalizing factor, Pi is the probability of movement in the i-th direction. New direction 0 is selected using the roulette method based on Pi probabilites.
3.4.2 Movement speed
Movement speed is deterministic and is defined by local conditions. If the
current location is not favorable, the individual moves quickly and if local
conditions are good it moves at a slower rate. The movement speed is defined by
logistic function:
(8) s = Smax{1/[1 + exp (CAl ))}
where Smax is maximum speed, Z is the model parameter, AL is a local attraction of the current location.
The attraction/repulsion for different factors Ak are combined into a value of local attraction AL in the following manner:
(9) Al =± WkAk
k=1
where wk (weight factors) are model parameters. Here s = 0 at AL s = Sm
at Al -ro.
3.4.3 Attraction/repulsion factors
The attraction/repulsion for each factor are represented in the following way:
Food quantity:
(10) Ai ,v = cvVj( dv + V)
where cv , dv are parameters, Vi is forage biomass density at i-th cell.
3.5 Energy balance equations
Dynamics of individual properties is determined based on energy balance equations.
3.5.1 Energy consumption Basic energy expenses
Basal metabolic rate for herbivores scales to body weight (11) BMR = cW075
where W is body mass in kg, estimation for c - 292.88 kJ (70 cCal) per day (see [38]).
Wild ruminants spend most of the day (>90%) foraging, resting/ruminating or walking between bedding and feeding sites. Thus, energy budgets can be easily calculated from activity budgets if the incremental costs of standing, traveling, and foraging are known. Work [42] contains estimations of the following costs for Moose:
Activity Cost relative to BMR (%)
Rest 1.17
Standing 25
Winter foraging 64.4
Summer foraging 51.5
Bedding 8.1
Ruminating 16.4
Table 1. Costs of Moose activities relative to BMR.
In this implementation we are not keeping track of the time budget of individuals and we assume that about half of the day is spent eating. If we add some time for rumination and other activities we get a daily energetic cost of (12) Ed = 445 x W075
Energy cost of locomotion
Locomotion costs can be expressed as energy expenditures of moving 1 kg of body mass over 1 kilometer (kJ/kg * km). Work [41] contains measurements of locomotion costs for elks of different ages, similar results obtained in [47]:
(13) E(n0__) = 12.43 x W066 x 5 x At
Additional energy costs related to movement in slopes. Their estimations can be found at [41], [46], [43]. We will not take into account down slope motion costs, and will calculate elevation difference as maximum elevation difference during movement:
(14) Eh = Ah x 23.97 xW
Where Eh - slope movement energy cost, Ah - maximum elevation difference
(km).
3.5.2 Energy gains
Individual gains energy as a result of foraging process. The amount of food consumed depends on a number of factors.
3.5.2.1 Seasonal appetite and potential growth.
Food consumption during winter is reduced even when animals have unlimited
supply of food. In [38] the following functions to estimate seasonal appetite and
potential growth for wapiti females as a function of day of the year are proposed:
'CYCL x1.4 + 0.0045 xW' max{0, Wmax - W} _
(16) CYCL = 0.6 + 0.4cos (6.186( julday -182)/365)
where CYCL - seasonal appetite (from 0 to 1), julday - julian date. We use PG values to calculate on of the restrictions for food consumption (sell below), and prevent unrealistic individual growth. The potential growth rate for a particular day of the year cannot exceed seasonal constraints or maximum weight, Wmax = 365 kg.
(15) PG = min
3.5.2.2 Energy from food.
The gross energy in browse and grass tissues is in the order of 18.5 kJ/g ([39], [43]), while the fraction that can be effectively processed by the animals (digestibility, dig) throughout the seasons, between 0.35 to 0.7 for grasses, between 0.35 and 0.57 for browse [40], and 0.55 - 0.7 for forbs [42].
Works [40], [37] used a digestibility curve for browse. [37] have digestibility of 50% until Jan 5 and then decreased linearly to 45% in April 15 where it remained constant until the beginning of the growing season.
Finally, metabolizable energy is 82% of digestible energy, and we get
(17) Ef = 18.5 x 0.82 x dig x F
where Ef is energy from food, F is food consumed (grams). F is defined by equation (24).
Digestibility is function of NDF:
(18) dig = (97 - 0.48NDF) x 0.01
3.5.2.3 Catabolism and anabolism.
A common cause for elk death is starvation. It is generally assumed that an animal will die if its body fat is below 1% of lean body mass ([48], [40]). Many physiological processes can be related to body mass. We represent individual body mass in kg by the state variable W, which is the sum of lean weight (M) and body fat (Fat). Body fat is represented by a separate variable Fat, also in kg. Fat mass can be up to 25% of body mass ([40], [37]).
If difference between energy cost and energy gained is positive, energy is fixed as protein and fat (anabolism). fat and protein are catabolized to meet the energy demand. Fat and protein ratio on catabolism and anabolism processes are 5:1 [40], fat has energy content 3812 kJ/g, protein - 2264 kJ/g [43]. Food consumption
Individuals try to consume enough food to meet their energetic demands and potential growth. The amount of food consumed is however kept within the limits
imposed by forage density, time available for foraging and digestive constraints. Daily voluntary intake (DVI, grams) is a good proxy for the maximum food ingestion limit set by digestive constrains and we use the allometric equation reported by Wilmshurst [49]:
(19) DVI = 2.5 - 0.049 NDF + 0.061 W09
Percent NDF is set as the weighted average of NDF in the cells visited by individual.
Short term intake rate (in grams per minute) can be represented as
(20) I = ^xV/(v + V )
where Rmax is a maximum intake rate (g/min), V forage biomass density, averaged for cells visited, and v is so-called half-saturation constant. For maximum intake rate, we used allometric function [45] fitted to a variety of mammalian herbivores:
0 71 2
Rmax = 0.45 W Estimations for v are quite variable, from 71.2 to 160.4 g/ m for elk [50].
The amount of food ingested (in grams of forage) is calculated as feeding time (minutes) times intake rate. Daily ingestion is always kept within the limit set by DVI. Maximum foraging time is set to 10 hours. Furthermore, ingestion should not exceed the amount of energy needed to reach the corresponding potential growth (Epg):
PGt
(21) Epg =---—---= PG x17755
0.8/38120 + 0.2 x 4/22640
where 0.8 is a proportion of energy fixed as fat (energy content is 38120 kJ/kg), 0.2 is a proportion of energy fixed as protein (energy content is 22640 kJ/kg). The amount of plant biomass needed to achieve the potential growth is
(22) C = (Epg + Es //(18.5 x 0.82 x dig)
where Es is energy spent during the day:
(23) Es = Ed + % + Eh
where Ed, El and Eh are calculated using equations (12), (13),(14) respectively. Finally, we calculate food consumed as
(24) F = min
(DVI x1000^ I x 600
C
Plant biomass dynamics
Taking into account fat and protein transformation processes elk biomass dynamics is defined by equations:
Fat(t +1) = Fat(t) + 0.8(Ef -Es)/38120 (25) M(t +1) = M(t) + 0.2(Ef - Es)/22640 W (t) = M (t) + Fat (t)
Ef and Es values define increase or decrease of body mass. If fat mass achieves 0 value, elk dies.
3.5.3 Plant biomass dynamics
Plant biomass dynamics is formed by two processes: elk foraging, and plants natural growth. Growth is defined by equation:
(26) V (t +1) = V (t) +
u
i,0
( V. ^ 1 —iv m j
Consumption by elk is defined by equation: (27) V (t +1) = Vl (t)--n = V (t)
nM2' ZV; (t) This is the sum for all cells visited by elk.
4. Modeling results
4.1 Data
We used experimental data obtained in Banff National Park landscapes (Rocky Mountains, Alberta, Canada).
4.1.1 Landscape data
As model landscapes we used experimental data from 2 real landscapes
(Landscape 1, Landscape 2), with 629 km area (landscape factors, forage), based on
GIS-mapping, with spatial resolution 250 m. See map with two landscapes marked on Picture 1.
Picture 1. Map with marked landcapes.
Landscapes presentation using developed application software, realizing model described, is shown on Pictures 2, 3 (brighter cells correspond to more forage density). For investigation purposes we divided each landscape for two equal parts.
Picture 2. Landscape 1 (model presentation).
Picture 3. Landscape 2 (modelpresentation).
See landscapes properties in table below:
Property Landscape 1 Landscape 2
Area, km 629 629
Elevation, m 1000-1200 1600-1800
Cell linear size, m 250 250
Forage density, r/ m 83,1 78,8
Table 2. Landscape properties. 4.1.2 Elks data
We compared modeling results with movement data of real elks (7 species) in this region, that was fixed during one year with 4 time interval, spatial resolution 30 m. For our purposes we converted data to 1 day interval.
4.2 Modeling results
The goal of our modeling was to obtain the distributions of daily distances and
turning angle of elks.
Modeling procedure
1. We model the movement of elks (20 species) in landscape for 1 year.
2. Based on these calculations we find model distributions: daily movement distances, daily turning angles.
3. We compare the model distributions with experimental ones. The quality criterion is the closeness of distributions defined by mean squared deviation of distribution density functions (obtained from model and experimental).
4. We change the model parameters to increase closeness of model and experimental distributions. The procedure is repeated until satisfactory result obtained.
We used two factors influencing selection of movement direction:
1. Forage biomass density
2. Tendency to keep movement direction
Key equations defining described distributions are (10), (8), (5). Table 2 contains parameters included in above mentioned equations; they were changed during the modeling procedure (free parameters):
Parameter Remarks
Forage density attraction factors (equation (10))
d n
s max Movement speed (equation (8))
z
PcF Tendency to keep movement direction (equation (5))
PcR
Table 3. Free model parameters values
Each of landscapes (Landscape 1, Landscape 2) was divided into two equal parts (Picture 2, 3). We applied modeling procedure first for Part 1, and obtained values of model parameters giving best modeling results. Then we calculated distributions for Part 2 using these values. This means we used Part 1 to tune the model, and Part 2 to check its consistency.
4.2.1 Experiment 1 (Landscape 1)
Values of parameters obtained:
Parameter Value Remarks
cn 1,0 Forage density attraction factors (equation (10))
d n 300,0
s max 13,0 Movement speed (equation (8))
8,4
PcF 0,5 Tendency to keep movement direction (equation (5))
PcR 0,95
Table 4. Landscape 1. Model parameters values.
See modeling results for two parts on Pictures 4-7. Mean squared deviations are shown in table below:
Distribution Part 1 Part 2
Distance 3,8 7,1
Turning angle 1,2 1,2
Table 5. Landscape 1. Mean squared deviation values.
M
Picture 4. Landscape 1, Part 1. Distances distribution (1 day)
25 1% 20 o
15
10
5 — linn
1 ■ 1 1 1 1 rl 1 ■ . ■ _J
250 1250 22. □ Mo 50 qenb ■ OKcnepuiv 3250 4250 M ieHT
Picture 6. Landscape 1, Part 2. Distances distribution (1 day)
o
-180
O Moflenb —■—Экcпepммeнт
4.2.2 Experiment 2 (Landscape 2)
Values of parameters obtained:
Parameter Value Remarks
S 1,0 Forage density attraction factors (equation (10))
d n 300,0
s max 13,0 Movement speed (equation (8))
t 9,3
PcF 0,5 Tendency to keep movement direction (equation (5))
PcR 0,95
Table 6. Landscape 2. Free model parameters values.
See modeling results for two parts on Pictures 8-11. Mean squared deviations are shown in table below:
Distribution Part 1 Part 2
Distance 3,8 5,1
Turning angle 1,1 1,1
Table 7. Landscape 2. Mean squared deviation values.
25 1% 20 o
15 -10 -5 n
— I | n n
III III! 1 ■ ■ ■ rm _■ - J
250 1250 □ 2250 Moflenb ■ OKcnepuMe 3250 4250 M iHT
Picture 8. Landscape 2, Part 1. Distances distribution (1 day)
-180 °
O Moflenb —■—Экcпepммeнт
25 1 20 -15 -10 -5 %
1 1 n l n
r r Mil 1111 1 ■ ■ ■ ™ ■ __ rl
250 1250 225 □ Mog 0 3, lerib ■ OKcnepuMeHT 250 4250 M
Picture 10. Landscape 2, Part 2. Distances distribution (1 day)
-180 °
O Mogenb —■—Экcпepммeнт
4.2.3 Modeling results analysis
The analysis of results shows the satisfactory correspondence of model and
experimental distributions.
For both landscapes the quality for turning angles distributions was higher then for distance distributions. It is also equal for both parts of landscape. The movement distances distribution quality is much worse. We can find the existence of second maximum in distribution (it is not monotonous). The quality of distribution for Part 2 is worse then for Part 1.
We can name following possible reasons for divergence:
1. Experimental data include movement of elks in wider landscape than modeled ones. Modeled landscapes were chosen as significantly different in conditions (different elevations and forage densities).
2. We used deterministic function to calculate daily distance (8), and it is not confirmed by any publications. This leads to the fact that distance distributions highly depends on forage density distributions, and this led to appearance of second maximum and lower quality of model distributions.
Based of this analysis we can propose further model improvement:
1. To conduct modeling for wider areas, and investigate in more detail the elk's movement data to find relations between distances and environmental factors.
2. To use stochastic mechanism for distance instead of deterministic one (8), with distribution depending on local environmental conditions. Taking into account the experimental distribution, we propose the distribution defined by equation (22) for distance. It is Poisson distribution with mean value depending on local attraction:
(22) f (s) = A exp(-As), A = S _ [1 + exp(ZA)]
The model and modeling results are used for further development of research conducted in University of Connecticut. Acknowledgements
We would want to thank Peter V. Turchin, Vladimir G. Redko for valuable recommendations on model improvement and calculations data processing, and also for useful comments on manuscript. This work was supported by National Science Foundation grant 0078130.
REFERENCES
[1] Sorokin P.A. The Categorization of Individual-Based Modeling. Electronic journal "Investigated in Russia" (in Russian), 2003, 50, c. 574-588.
http://zhurnal.ape.relarn.ru/articles/2003/050.pdf
[2] Underwood H.B., Porter W.F. Values and science: white-tailed deer management in eastern national parks. Transactions 56th North American Wildlife and Natural Resource Conference, Edmonton, Alberta, Canada, 1991, pp. 67-73.
[3] Risenhoover K.L., Underwood H.B., Yan W., Cooke J.L. A spatially-explicit modelling environment for evaluating deer management strategies. 366-379 in: W.J. McShea, H.B. Underwood, and J. Rappole, eds., The science of overabundance: deer ecology and population management. Washington, DC, USA: Smithsonian Institution Press, 1997.
[4] Abbott C.A., Berry M.W., Comiskey E.J., Gross L.J., Luh H.-K. Computational Models of White-Tailed Deer in the Florida Everglades. IEEE Computational Science and Engineering, 1997, 4(4).
[5] Abbott, C. A. A Parallel Individual-Based Model of White-Tailed Deer in the Florida Everglades. M.S. Thesis. University of Tennessee, Department of Computer Science, 1995.
[6] Comiskey E. J., Gross L. J., Fleming D. M., Huston M. A., Bass O. L., Luh H.-K., Wu Y. A spatially-explicit individual-based simulation model for Florida panther and white-tailed deer in the Everglades and Big Cypress landscapes. Proceedings of the Florida Panther Conference, Ft. Myers Fla., Nov. 1-3, 1994, Dennis Jordan, ed., U.S. Fish and Wildlife Service, pp. 494-503.
[7] Thomas J. W., Toweill D. E.. Elk of North America: ecology and management. Stackpole Books, Harrisburg, Pennsylvania, 1982, 698 pp.
[8] Albon S.D., Langvatn R. Plant phenology and the benefits of migration in a temperate ungulate. Oikos, 65, 1992, pp. 502-513.
[9] Brazda, A. R. Elk migration patterns, and some of the factors affecting movements in the Gallatin river drainage, Montana. J. of Wildlife Management, 17, 1953, pp. 9-23.
[10] Langvatn R., Hanley T.A. Feeding-patch choice by red deer in relation to foraging efficiency; An experiment. Oecologia, 95, 1993, pp. 164-170.
[11] Wilmshurst J.F., Fryxell J.M., Hudson R.J. Forage quality and patch choice by wapiti (Cervus elaphus). Behav. Ecol. 6, 1995, pp. 209-217.
[12] Staines B. W. A review of factors affecting deer dispersion and their relevance to management. Mammal Review 4, 1974, pp. 79-91.
[13] Bowyer R.T. Activity, movement, and distribution of Roosevelt elk during rut. Journal of Mammalogy 62, 1981, pp. 574-582.
[14] Edge W. D., Marcum C. L. Movements of elk in relation to logging disturbances. Journal of Wildlife Management 49, 1985, pp. 926-930.
[15] Edge W. D., Marcum C. L., Olson S. L., Lehmkuhl J. F. Nonmigratory cow elk herd ranges as management units. Journal of Wildlife Management, 50, 1986, pp. 660-663.
[16] Cole E.K., Pope M.D., Anthony R.G. Effects of Road Management on Movements and Survival of Roosevelt Elk. Journal of Wildlife Management, v. 61, № 4, 1997.
[17] WallisDeVries M.F. Habitat quality and the performance of large herbivores. Grazing and Conservation Management. Kluwer Academic Publishers, London, 1998, pp. 275-320.
[18] Etzenhouser M. J., Owens M. K., Spalinger D. E., Murden S. B. Foraging behavior of browsing ruminants in a heterogeneous landscape. Landscape Ecology 13, 1998, pp. 55-64.
[19] Hobbs N. T., Baker D. L., Bear G. D., Bowden D. C. Ungulate grazing in sagebrush grassland: mechanisms of resource competition. Ecological Applications, 6, 1996, pp. 200-217.
[20] Pastor J. B., Moen R., Cohen Y. Spatial heterogeneities, carrying capacity, and feedback patterns in animal landscape interactions. The Journal of Mammalogy, 78, 1997, pp. 1040-1052.
[21] Kareiva P. Population dynamics in spatially complex environments: theory and data. Philosophical Transactions of the Royal Society of London B 330, 1990, pp. 175-190.
[22] Turchin P. Translating foraging movements in heterogeneous environments into the spatial distribution of foragers. Ecology, 72, 1991, pp. 1253-1266.
[23] Turchin P., Reeve J. D., Cronin J.T., Wilkens R.T. Spatial pattern formation in ecological systems: bridging theoretical and empirical approaches, 1997, pp. 195210 in: Bascompte J, Solé RV (eds). Modelling Spatiotemporal Dynamics in Ecology. Landes Bioscience, Austin, TX.
[24] Turchin P. Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer Associates, Sunderland, MA, 1998.
[25] Levin S.A. The problem of pattern and scale in ecology. Ecology 73, 1992, pp. 1943-1967.
[26] Hanski I., Simberloff D. The metapopulation approach, its history, conceptual domain and application to conservation, 1997, pp. 5-26 in I. Hanski and M. Gilpin, editors. Metapopulation biology: ecology, genetics and evolution. Academic Press, London, UK.
[27] Turner M.G., Pearson S.M., Romme W.H., Wallace L.L. Landscape heterogeneity and ungulate dynamics: what spatial scales are important? 1997, pp. 331-348 In: J.A. Bissonette (ed.). Wildlife and landscape ecology. Springer-Verlag, New York.
[28] Wiens J. A. Metapopulation dynamics and landscape ecology, 1997, pp. 4362 in I. Hanski and M. Gilpin., editors. Metapopulation biology: ecology, genetics and evolution. Academic Press, San Diego, California, USA.
[29] Okubo A., Levin S. A. Diffusion and ecological problems: modern perspectives, second edition, Springer-Verlag, 2001.
[30] Okubo A. Diffusion and Ecological Problems: Mathematical Models. N.Y. etc.: Springer-Verlag, 1980, 254 p. (Biomathematics; Vol. 10).
[31] Viswanathan G. M., Afanasyev V., Buldyrev S. V., Murphy E. J., Prince P. A., Stanley H. E. Levy flight search patterns of wandering albatrosses. Nature, 381, 1996, pp. 413-415.
[32] Johnson C. J., Parker K. L., Heard D.C., Gillingham M. P. Movement parameters of ungulates and scale-specific responses to the environment. Journal of Animal Ecology 71, 2002, pp. 225-235.
[33] Viswanathan G. M., Bartumeus F., Buldyrev S. V., Catalan J., Fulco U. L., Havlin S., da Luz M. G. E., Lyra M. L., Raposo E. P., Stanley H. E. Levy flight random searches in biological phenomena. Physica a-Statistical Mechanics and Its Applications 314, 2002, pp. 208-213.
[34] Nams V. O. The VFractal: A new estimator for fractal dimension of animal movement paths. Landscape Ecology 11, 1996, pp. 289-297.
[35] Fritz H., Said S., Weimerskirch H. Scale-dependent hierarchical adjustments of movement patterns in a long-range foraging seabird. Proceedings of the Royal Society of London Series B-Biological Sciences 270, 2003, pp. 1143-1148.
[36] Fauchald P., Tveraa T. Using first-passage time in the analysis of arearestricted search and habitat selection. Ecology 84, 2003, pp. 282-288.
[37] Delgiudice, G. D., R. A. Moen, F. J. Singer, and M. R. Riggs. 2001. Winter nutritional restriction and simulated body condition of Yellowstone elk and bison before and after the fires of 1988. Wildlife Monographs:1-60.
[38] Hudson, R. J., and R. J. Christopherson. 1985. Maintenance metabolism. Pages 121-142 in R. J. Hudson and R. G. White, editors. Bioenergetics of Wild Herbivores. CRC Press, Boca Raton, Fla.
[39] Hjeljord, O., F. Sundstol, and H. Haagenrud. 1982. The Nutritional Value of Browse to Moose Alces-Alces. Journal of Wildlife Management 46:333-343.
[40] Moen, R., J. Pastor, and Y. Cohen. 1997. A spatially explicit model of moose foraging and energetics. Ecology 78:505-521.
[41] Parker, K. L., C. T. Robbins, and T. A. Hanley. 1984. Energy expenditures for locomotion by mule deer and elk [Odocioleus hemionus, Cervus elaphus nelsoni]. Jounral of Wildlife Management 48:474-488.
[42] Renecker, L. A., and R. J. Hudson. 1988. Seasonal quality of forages used by moose in the aspen-dominated boreal forest, central Alberta. Holarctic Ecology 11:111-118.
[43] Robbins, C. T. 1993. Wildlife feeding and nutrition, second edition. Academic Press, San Diego.
[44] Robbins, C. T., R. S. Podbielancik-Norman, D. L. Wilson, and E. D. Mould. 1981. Growth and nutrient consumption of elk calves compared to other ungulate species. Journal of Wildlife Management 45:172-186.
[45] Shipley, L., J. E. Gross, D. E. Spalinger, N. T. Hobbs, and B. A. Wunder. 1994. The scaling of intake rate in mammalian herbivores. American Naturalist 143:1055-1082.
[46] Taylor, C. R., S. L. Caldwell, and R. V.J. 1972. Running Up and Down Hills: Some Consequences of Size. Science 178:1096-1097.
[47] Taylor, C. R., and N. C. Heglund. 1982. Energetics and mechanics of terrestrial locomotion. Annual Review of Physiology 44:97-107.
[48] Turner, M. G., Y. Wu, L. L. Wallace, W. H. Romme, and A. Brenkert. 1994. Simulating winter interactions among ungulates, vegetation, and fire in northern Yellowstone Park. Ecological Applications 4:472-496.
[49] Wilmshurst, J. F., J. M. Fryxell, and C. M. Bergman. 2000. The allometry of patch selection in ruminants. Proc R Soc Lond, Biol sci 267:345-349.
[50] Wilmshurst, J. F., J. M. Fryxell, and R. J. Hudson. 1995. Forage quality and patch choice by wapiti (Cervus elaphus).
